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Identical expressions
e^(2sinx/ four)
e to the power of (2 sinus of x divide by 4)
e to the power of (2 sinus of x divide by four)
e(2sinx/4)
e2sinx/4
e^2sinx/4
e^(2sinx divide by 4)

The derivative of the function/ e^(2sinx/4)

Derivative of e^(2sinx/4)

The solution

You have entered [src]

 2*sin(x)
 --------
    4    
E

$$e^{\frac{2 \sin{\left(x \right)}}{4}}$$

E^((2*sin(x))/4)

Detail solution

Let $u = \frac{2 \sin{\left(x \right)}}{4}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 \sin{\left(x \right)}}{4}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $2 \cos{\left(x \right)}$
  So, the result is: $\frac{\cos{\left(x \right)}}{2}$
The result of the chain rule is:

$\frac{e^{\frac{2 \sin{\left(x \right)}}{4}} \cos{\left(x \right)}}{2}$
Now simplify:

$\frac{e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{2}$

The answer is:

$\frac{e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        2*sin(x)
        --------
           4    
cos(x)*e        
----------------
       2

$$\frac{e^{\frac{2 \sin{\left(x \right)}}{4}} \cos{\left(x \right)}}{2}$$

Simplify

The second derivative [src]

                      sin(x)
                      ------
/   2              \    2   
\cos (x) - 2*sin(x)/*e      
----------------------------
             4

$$\frac{\left(- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\frac{\sin{\left(x \right)}}{2}}}{4}$$

Simplify

The third derivative [src]

                                  sin(x)
                                  ------
/        2              \           2   
\-4 + cos (x) - 6*sin(x)/*cos(x)*e      
----------------------------------------
                   8

$$\frac{\left(- 6 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - 4\right) e^{\frac{\sin{\left(x \right)}}{2}} \cos{\left(x \right)}}{8}$$

Simplify

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Derivative of e^(2sinx/4)

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The solution